{"paper":{"title":"A dichotomy theorem for the generalized Baire space and elementary embeddability at uncountable cardinals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Dorottya Szir\\'aki, Jouko V\\\"a\\\"an\\\"anen","submitted_at":"2015-08-22T18:57:39Z","abstract_excerpt":"We consider the following dichotomy for $\\Sigma^0_2$ finitary relations $R$ on analytic subsets of the generalized Baire space for $\\kappa$: either all $R$-independent sets are of size at most $\\kappa$, or there is a $\\kappa$-perfect $R$-independent set. This dichotomy is the uncountable version of a result found in (W. Kubi\\'s, Proc. Amer. Math. Soc. 131 (2003), no 2.:619--623) and in (S. Shelah, Fund. Math. 159 (1999), no. 1:1--50). We prove that the above statement holds assuming $\\Diamond_\\kappa$ and the set theoretical hypothesis $I^-(\\kappa)$, which is the modification of the hypothesis "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.05539","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}